Atomistic Modeling of Processing Issues for Transparent Polycrystalline Alumina

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POUR L'OBTENTION DU GRADE DE DOCTEUR ÈS SCIENCES

acceptée sur proposition du jury: Prof. K. Scrivener, présidente du jury

Dr P. Bowen, directeur de thèse Dr U. Aschauer, rapporteur Prof. N. Marzari, rapporteur Prof. S. Parker, rapporteur

Atomistic Modeling of Processing Issues for Transparent

Polycrystalline Alumina

THÈSE NO_{6019 (2013)}

ÉCOLE POLYTECHNIQUE FÉDÉRALE DE LAUSANNE PRÉSENTÉE LE 29 NOVEMBRE 2013

À LA FACULTÉ DES SCIENCES ET TECHNIQUES DE L'INGÉNIEUR LABORATOIRE DE TECHNOLOGIE DES POUDRES

PROGRAMME DOCTORAL EN SCIENCE ET GÉNIE DES MATÉRIAUX

Suisse 2013 PAR

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Atomistic Modeling of Transparent Alumina

A

cknowledgements

xq:czZãk xq:foZ‘.kw xq:nsZoks egs“oj%A

xq: lk{kkr~ ijczZã rLeS Jh xq#os ue%AA

(The Guru is Brahma (the creator), Lord Vishnu (the preserver), and Lord Shiva (the desroyer). To that very Guru I bow, for He is the Supreme Being, right before my eyes.)

First of all, I would like to thank my thesis director and guru Dr. Paul Bowen. He has been a great source of motivation and encouragement during the course of my entire thesis. I cherish the freedom of experimentation and flexibility of working, which I got with him. I always enjoyed his pragmatic approach towards science as well as numerous interesting discussions we had on sports and life in general. I am also thankful to him for supporting me to go to various conferences and to collaborate with different groups.

I would also like to thank my thesis jury members for examining my thesis and giving useful feedback and comments to improve my thesis and gain further insights into my work.

I would also like to thank my colleague and friend Sandra Galmarini for being a continuous support during my thesis. Critical discussion with her always helped me refine my work. It was a pleasure to work with Dr. Uli Aschauer on various topics during my PhD thesis, who helped me learning new simulation methods. I would also like to thank Prof. Steve Parker for welcoming me at University of Bath and getting me started with general grain boundary calculations.

I am also grateful to Prof. Hofmann and Prof. Jacques Lemaitre for welcoming me at LTP and providing a very conducive environment for research at LTP, and our lab secretary Ruth Gacoin who made the administrative work much easier.

The experience of the PhD would have been much lesser if not for the wonderful friends and colleagues I met at LTP. Arthur, Henning, Lionel, Gaby, Kat, June, Vianney, Piyak, Vanessa, Goldy, Saso, Sandra all made my PhD stay at LTP more enriching and joyous. I think numerous discussions over lunch and coffee breaks ranging from science, economics and politics to society and religion have helped me broaden my outlook about life in general.

I also consider myself fortunate to have wonderful friends around me. Big thanks to my best friends Paula, Nishanth and Kat for being there to share my life in happy as well as depressing moments. I have learnt lots of important lessons to face the

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situations in life thanks to them. Ram, Suri, Soni, Devika, Swati are few other names, which I can not forget to mention here. Being so far away from home, it was very nice to have such a vibrant and active Indian community at EPFL. The events and festivals celebrated by Indian association at EPFL made us miss the home less on those important occasions.

I would also like to thank my spiritual guru Shri Ganesh Bagaria for initiating the curiosity within me to understand myself better. The knowledge I have gained from him has helped me set my priorities in life straight and better understanding of human values and relationships.

Last but not the least by any means, I would like to express my highest gratitude to my parents and my family. The hardships and difficulties my late father and my mother went through to provide me the best possible education is the reason that I am where I am today. I can never thank them enough for their unconditional love and support. The kind of confidence and belief my family had in me has always pushed me to do my best in my endeavor to be a better human being.

I am sure there might be many more people to thank. I am sorry if I have forgotten to mention them here.

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A

bstract

Transparent polycrystalline alumina has many possible promising areas of application from jewelry and the watch industry to wave guides, energy economical lamp envelopes, and optical windows. Ultrahigh density, submicron sized grains and/or oriented microstructures have been identified as the key requirements to synthesize transparent alumina. The highest real inline transmittance (RIT) aluminas reported in the literature are still not good enough to be used for transparent applications.

The goal of the present thesis was to use atomistic modeling to understand the basic mechanisms of the physical/chemical phenomena involved in the various issues pertaining the processing of transparent alumina. The three main issues which were addressed in the present work are: segregation of cation-dopants/anion-impurities to the alumina interfaces, solid state oxygen diffusion in alumina, and adsorption of polymers on alumina surfaces.

Doping of alumina with transition elements (e.g. Y, Mg, La) has been used in the literature for grain growth reduction and creep enhancement. Codoping with a combination of dopants (e.g. Mg-La) has been reported to be more effective. However the atomistic level effects of codoping on alumina microstructure and hence on properties are not very well understood. The energy minimization method was used to calculate the segregation energies and the relaxed atomistic structures of as many as 9 codoped (Y-La, Mg-La, Mg-Y) surfaces and twin grain boundaries (GBs). Only codoping with a combination of bivalent-trivalent (Mg-La and Mg-Y) dopants was found to be energetically more favorable than single doping. Disparity in the ionic sizes was identified as the key reason for the favorable codoping with Mg. Effects of the dopants type and concentration on the GB atomistic structures have been discussed in the light of the GB complexion transitions and GB packing. Coordination number calculations were made to analyze the GB chemical environment.

The existence of anion impurities such as chlorides and sulphates in industrial alumina powder synthesis is well known. But its effects on the processing of alumina ceramics have been grossly neglected. Energy minimization calculations showed that the segregation of Cl is 4-6 times stronger than the cation dopants. Cl-Al coordination number analysis suggests strong adhesion of Cl on the powder surface, making the removal of Cl ions difficult at low temperatures.

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Oxygen diffusion plays an important role in grain growth and densification during the sintering of alumina ceramics and governs high temperature processes such as creep. The atomistic mechanism for oxygen diffusion in alumina is however still debated. The calculations are usually performed for perfectly pure crystals, whereas virtually every experimental alumina sample contains a significant fraction of impurity/dopants ions. In the present study atomistic defect cluster and nudged elastic band calculations have been used to model the effect of Mg impurities/dopants on defect binding energies and migration barriers. It was found that oxygen vacancies can form energetically favorable clusters with Mg, which reduces the number of mobile species. Moreover diffusive jumps leading away from Mg have migration energies up to twice the value in pure alumina, whereas those approaching Mg are lowered by up to a factor of four, which will slow down the kinetics of diffusion. Other effects of Mg such as vacancy destabilization and the vacancy-vacancy interactions have also been discussed in detail.

Majority of the computational segregation studies are done on the highly symmetrical twin grain boundaries. However, the fraction of special twin grain boundaries found in sintered alumina samples is reported to be very small. Therefore, to fulfill the ultimate goal of the simulations, i.e. linking the simulations with the experiments, experimentally characterized general GBs were simulated using near coincidence GB approach and the energy minimization method. Although the segregation of Y was found to be energetically favorable, dopants were found to be occupying only 25% cation sites on the GB. GB complexion phases, which are less favorable to grain growth reduction, were found to be more probable on the general GBs in contrast to the twin GBs.

Controlling the agglomeration of ultrafine powders is a big challenge in the processing of nano scaled ceramics. Understanding of the conformation of adsorbed dispersants and the interplay of the adsorption with powder surface characteristics is still limited and requires further work on a rather fundamental level. The present thesis could not address this issue in detail due to the unavailability of an adequate force field. The preliminary results on the development of such a force field as well as the progress made so far are discussed in the last chapter of the thesis.

The present thesis helps understand basic fundamental issues pertaining to the processing-microstructure-property relationship in transparent alumina which should help overcome the major roadblocks in the progress of the field of transparent alumina ceramics. The work is generic and the methods can be successfully applied to other ceramic systems.

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R

ésumé

L’Alumine polycristalline transparente a de nombreux domaines d'application possibles prometteurs allant des bijoux à l'industrie horlogère en passant par les guides d'ondes, les enveloppes des lampes à économie d’énergie ou les fenêtres optiques. L’ultra densité, les grains de tailles submicroniques et / ou les microstructures orientées ont été identifiés comme les besoins principaux pour synthétiser de l’alumine transparente. Les plus hautes valeurs de transmittance (RIT) des alumines rapportées dans la littérature ne sont pas encore assez bonnes pour être utilisé pour des applications transparentes.

L'objectif de cette thèse a donc été d'utiliser la modélisation atomistique pour comprendre les mécanismes de base des phénomènes physicochimiques impliqués dans les diverses questions relatives au traitement de l'alumine transparente. Les trois principales questions qui ont été abordées dans le présent ouvrage sont: la ségrégation des dopants-cationiques/impuretés-anioniques aux interfaces de l’alumine, l’état de diffusion de l'oxygène dans l'alumine à l’état solide et l'adsorption des polymères sur des surfaces d'alumine.

Le dopage de l'alumine avec des éléments de transition (par exemple Y , Mg, La) sont utilisés pour la réduction de la croissance des grains et de l'amélioration du fluage . Le co-dopage avec une combinaison de dopants (par exemple Mg- La) semble être la plus efficace. Toutefois, les effets des co-dopages, au niveau atomistique, sur la microstructure d'alumine et donc sur ses propriétés ne sont pas très bien comprises. La méthode de minimisation de l'énergie a été utilisée pour calculer les énergies de ségrégation et les structures atomiques détendues de 9 surfaces co-dopées (Y -La, Mg -La, Mg -Y) et de joints de grains miroirs (GBs). Le co-dopage avec seulement des combinaisons de dopants bivalents ou trivalents (Mg -La et Mg -Y) a démontré être énergétiquement plus favorable que le dopage unique. La raison principale de l’efficacité d’un co-dopage au Mg semble être la disparité des tailles ioniques. Les effets du type de dopants et de leur concentration sur les structures atomiques ont été discutés par les transitions de complexion et de compactage des GBs. Des calculs de nombres de coordination analysèrent l'environnement chimique des GBs.

L'existence d'impuretés anioniques telles que les chlorures et les sulfates dans la synthèse industrielle de poudre d'alumine est bien connue. Mais ses effets sur le traitement des céramiques d'alumine sont mal compris. Des calculs de minimisation de l'énergie ont montré que la ségrégation des Cl est 4 à 6 fois plus forte que les dopages cationiques. L’analyse du nombre de coordination Cl - Al suggère une forte adhérence des Cl sur la surface de la poudre, rendant leur élimination difficile.

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La diffusion de l'oxygène joue un rôle important dans la croissance et la densification des grains lors du frittage des alumines régissant les processus à haute température tel que le fluage. Le mécanisme atomistique de diffusion de l'oxygène dans l'alumine est cependant toujours débattu. Les calculs sont généralement effectués pour des cristaux parfaitement purs, alors que pratiquement tous les échantillons expérimentaux d'alumine contiennent une fraction importante d'impuretés ou d’ions de dopants. Dans cette étude, des clusters de défaut atomiques et des calculs sur les liaisons élastiques ont été utilisées pour modéliser l'effet des impuretés de Mg / dopants sur les défauts d’énergies de liaison et les obstacles de migration. Il a été constaté que les lacunes d'oxygène peuvent former avec le Mg des amas énergétiquement favorables réduisant le nombre d'espèces mobiles. De plus les sauts diffusants s’éloignant du Mg ont des énergies de migration deux fois plus grande que pour l'alumine pure, quand ceux s’approchant du Mg sont diminués d’un facteur quatre, ralentissant leur cinétique. D’autres effets du Mg comme la déstabilisation de lacunes et les interactions lacune - inoccupation ont été examinés. La majorité des calculs des études de ségrégation sont effectués sur les limites symétriques de grains miroirs. Cependant, la fraction des joints de grains miroirs spéciaux dans l'alumine frittée est très faible. Par conséquent, pour atteindre l'objectif final de la simulation (relier les simulations aux expériences) des GBs caractérisés expérimentalement ont été simulées en utilisant l'approche GB de coïncidence et la méthode de minimisation de l'énergie de proximité. Bien que la ségrégation des Y s'est avéré être énergétiquement favorable, seuls 25% des sites cationiques des GBs sont occupés. Leurs phases de complexion, moins favorables à la réduction de la croissance des grains, sont plus probables sur le GBs général que sur les miroirs.

Le contrôle de l'agglomération des poudres ultrafines est un grand défi dans le traitement des nanocéramiques. La compréhension de la conformation des dispersants adsorbées et de l'interaction de l'adsorption avec les propriétés de surface est encore limitée, nécessitant des études plus fondamentales. Dans cette thèse cette question n’a pas été traitée en détail due à l'indisponibilité d'un champ de force adéquat. Les résultats préliminaires et les progrès réalisés sur l'élaboration de ce champ de force sont toutefois abordés dans le dernier chapitre.

Cette thèse permet de comprendre les facteurs fondamentaux relatifs aux relations traitement, microstructure et propriété pour l’alumine transparente permettant ainsi d’aider à surmonter les principaux obstacles à l'avancement du domaine de la céramique d’alumine transparente. Ce travail étant générique, les méthodes peuvent également être appliquées avec succès à d'autres systèmes céramiques.

Mots Clefs: Modélisation atomistique, Alumine, ségrégation, défauts, impuretés,

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List of Figures

Figure 1.1. Surfaces, grain boundaries and pores are the sources of light scattering a polycrystalline material... 2 Figure 2.1. Schematic diagram of various types of bonded interaction potentials and their function form, (a) bonded potential, (b) angle potential, and (c) torsion potential ... 12 Figure 2.2. Functional form of two main types of non-bonded dispersion interactions, (a) Buckingham potential for Al-O and O-O pairs, and (b) typical shape of a Lenard-Jones potential ... 15 Figure 2.3. (a) Surface cut along the desired orientation of the slab surface, and (b) refilling of the left out atoms after the surface cut into the slab cell ... 18 Figure 2.4. 2-D periodic boundary conditions are applied in the surface plane of the slab and the repetition of the slab unit cell is done along the direction normal to the surface to construct a slab cell. ... 18 Figure 2.5. A surface cell consists of 2 regions. Atoms in the region 1 are allowed to relax, while the atoms in the region 2 are kept fixed during energy minimization. ... 19 Figure 2.6. A mirror twin grain boundary consists of 2 surfaces with the same miller index back to back and region 1 and region 2 in each block are similar as in the case of the surface cell. ... 20 Figure 2.7. Schematic represenation of a general grain boundary with different interplanar spacing in each grain for hexagonal systems. ... 22 Figure 2.8. In steepest decent method, the search direction is normal to the force, while the conjugate gradient method mixes the previous search direction with the direction of the force to speed up the convergence [44]. ... 25 Figure 2.9. Newton Raphson method for energy minimization uses both the force as well as the curvature of the energy surface to find the search direction. It finds the root of the equation (dE/dx=0). ... 26 Figure 2.10. Activation energy (ΔE) for a reaction is calculated using nudged elastic band method as the difference of the energy between the saddle point state and the initial state. ... 28 Figure 2.11. NEB is a method to find minimum energy path for the transition from a known initial state to a known final state. The diagrams shows the various forces acting on the elastic band during energy minimization of the elastic band. [49] ... 29

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Figure 2.12. Time evolution of the metadynamics potential. Thick solid line is the the starting potential and it is steadily filled with Gaussians (dotted lines). The dynamic evolution is labeled by the number of dynamical iterations. [51] ... 31 Figure 3.1. (a) single site substitution, (b) permutation of lowest energy sites for multiple dopant configurations, (c) codoping configuration as a combination of respective single dopant configurations ... 38 Figure 3.2. Comparison of probabilistic codoping with randomly chosen configurations, (a) Mg doped, (b) Y doped, and (c) La-Y doped ... 40 Figure 3.3. Representative segregation energy plots for La-Y codoped surfaces. The lines serve as visual guides only. ... 46 Figure 3.4. Representative segregation energy plots for La-Y codoped GB’s. The lines serve as visual guides only. ... 47 Figure 3.5. Representative segregation energy plots for Mg-La codoped a) (00.1) surface, b) (11.0) surface, c) (01.2) GB, d) (11.3) GB. Lines are just visual guides only. ... 50 Figure 3.6. Representative segregation energy plots for Mg-Y codoped, (a) (00.1) surface, (b) (11.3) surface, (c) (01.2) GB, (d) (11.1) GB. Lines are just visual guides. . 52 Figure 3.7. Codoping segregation energy comparison for three doping combinations for (a) (11.1) surface, and (b) (11.1) GB. The lines are visual guides only. ... 54 Figure 3.8. Interface structure dependent cosegregation and atomic arrangment of La-Y codoped α-alumina surfaces. All the atomistic structures are shown from the side view parallel to the grain boundary/surface plane with the white lines showing roughly the position of the grain boundary or surface plane. Oxygen ions are shown in red, aluminum ions in violet, Y in light blue and La in dark blue. ... 56 Figure 3.9. Interface structure dependent cosegregation and atomic arrangment of La-Y codoped α-alumina grain boundaries. All the atomistic structures are shown from the side view parallel to the grain boundary/surface plane with the white lines showing roughly the position of the grain boundary or surface plane. Oxygen ions are shown in red, aluminum ions in violet, Y in light blue and La in dark blue. ... 57 Figure 3.10. Complexion transition in La-Y codoped (11.1) grain boundary with the increasing dopant concentration. a) 1.93 at./nm2, b) 3.86 at./nm2, c) 5.79 at./nm2, d) 7.72 at./nm2, e) 9.65 at./nm2, f) 11.58 at./nm2, and g) 13.51 at./nm2. La is dark blue and Y is light blue. All the grain boundaries are viewed parallel to the GB plane. Line at the grain boundary is only a rough estimate of the grain boundary position. ... 59 Figure 3.11. Energetic probabilities of oxygen vacancy formation at the grain boundaries. The sites with smaller and lighter color atoms have higher probability and darkest and largest one have the least possibility. All the grain boundaries are viewed parallel to the GB plane. ... 60

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Figure 3.12. Formation of dopant atomic layer on Y-La codoped (01.2) grain boundary with increasing dopant concentration. Oxygen ions are shown in red, aluminum ions in violet, Y in light blue and La in dark blue. ... 61 Figure 3.13. Comparison of single doped and codoped (11.1) grain boundaries (a) Mg doped, (b) La doped, (c) Y- doped, (d) Mg-La codoped, (e) Mg-Y codoped, and (f) La-Y codoped (Concentration=7.74 at./nm2). Oxygen ions are shown in red, aluminum ions in violet, Y in light blue, La in dark blue and Mg in green. ... 61 Figure 3.14. (11.1) Y-La codoped surface looked from the top perpendicular to the surface. Specific coordinative arrangement/coupling effect is seen, where oxygen atom is always surrounded by two different types of dopant ions. ... 62 Figure 4.1. STEM-EDX spectroscopy results showing the presence of Cl impurities at a grain boundary and the triple points. ... 71 Figure 4.2. Representative segregation energy plots for Cl segregated (a) (01.2) surface, and (b) (11.1) surface. Lines are just visual guides. ... 72 Figure 4.3. Representative segregation energy plots for Cl segregated (a) (01.2) GB, and (b) (11.1) GB. Lines are just visual guides. ... 73 Figure 4.4. Atomistic structures of the interface at their respective characteristic concentrations as mentioned in the table 2, (a) (01.2), (b) (10.0), (c) (11.1) surfaces, and (d) (11.1), (e) (10.1), (f) (01.2) GB’s. All the structures are seen parallel to the interface plane. Al atoms are in pink, Oxygen in red and Cl atoms are in green. ... 74 Figure 4.5. Evolution of the atomistic structure of a (11.1) GB with the increasing concentration of Cl ions, (a) 2.8 at./nm2, (b) 5.6 at./nm2, (c) 8.3 at./nm2, (d) 11.11 at./nm2, and (e) 13.90 at./nm2. All the structures are seen parallel to the grain boundary plane. Al atoms are in pink, Oxygen in red and Cl atoms are in green. ... 75 Figure 5.1. Migration barrier of a class III jump was calculated for 5 supercell sizes. The migration barrier is seen to be converging for a 3x3x1 supercell. ... 83 Figure 5.2. NEB calculations were done with 10, 20 and 30 intermediate images. Activation energy is shown to be converging with 10 intermediate images. ... 83 Figure 5.3. Migration barriers for the three classes of primary jumps, computed using the nudged elastic band (NEB, solid lines) and metadynamics (MTD, dashed lines) methods. For NEB the whole minimum energy pathway is given, while for MTD only the height of the saddle point is indicated. ... 86 Figure 5.4. Binding energy of the VO and MgAl as a function of their distance

calculated by the Mott Littleton method. The blue line is a 1/d fit to the data. ... 88 Figure 5.5. Binding energy of the VO and MgAl as a function of their distance

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Figure 5.6. Binding energy per VO for defect clusters of one MgAl and one to four VO.

Negative values indicate energetically favorable binding. ... 89 Figure 5.7. Migration barriers for the diffusive jumps of oxygen vacancies as a function of the initial distance of the oxygen vacancy from Mg for primary jumps of (a) class I (2.42 Å), (b) class II (2.54 Å) and (c) class III (2.65 Å). Dashed lines are the migration barriers in pure alumina for the respective class of jumps. ... 90 Figure 5.8. Migration barriers of class III diffusive jumps as a function of the difference in the initial and final distance of the VO from MgAl. Negative values on the

x-axis represent jumps where VO approaches MgAl. The dashed line is the migration

barrier in pure alumina. ... 91 Figure 5.9. Migration barriers of three classes of jumps computed at two different c-axis lengths, keeping the potential parameters constant. ... 93 Figure 5.10. Migration barriers computed using the Lewis setup, compared to results obtained with the Binks potential and DFT calculations. ... 93 Figure 5.11. Change in migration barrier induced by the presence of Mg close to an oxygen vacancy. Results shown are for a small set of test cases. ... 94 Figure 5.12. Minimum energy pathways for jumps starting from vacancy positions, which became unstable due to the presence of MgAl. The inset shows a schematic

view of the relaxation process leading from the metastable (m) position to the new initial (i) position. Going from the new initial (i) position to the final (f) position now requires two diffusive jumps going via the intermediate metastable (m) state. This two-step nature is reflected by the two barriers observed in the minimum energy pathways (MEPs). The lines are cubic splines interpolated between NEB images. Red and blue solid curves are the MEPs of the highest barrier class II and class III jumps, respectively from this initial position. ... 95 Figure 5.13. Minimum energy pathways showing the effect of VO-VO interaction by

addition of a second VO at different nearest neighbor sites. The dashed horizontal

line is the migration barrier of the jump without a second VO. Pathways with a

destabilized initial position are shown in black, those with an increase in barrier in red, the one with an unaffected barrier in grey and the one with a lower barrier in blue. The lines are cubic splines interpolated between NEB images. ... 96 Figure 5.14. a) Mean square displacement versus time plots for oxygen atomic migration at different temperatures calculated using kinetic Monte Carlo simulations, b) Diffusion coefficients plots versus inverse of temperature to calculate macroscopic atomic diffusion law. ... 98 Figure 6.1. STEM-EDX spectroscopy results showing the presence of dopants and impurities at a grain boundary, (a) Aluminum, (b) Chlorine, (c) Lanthanum, and (d) Yttrium. ... 104

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Figure 6.2. Line scan quantitative EDX analysis of a GB showing the presence of La and Y in the narrow region of the GB. ... 105 Figure 6.3. Segregation energy for single yttrium dopant sites situated at different depths from the near coincidence GB with (00.1) and (01.3) as GB parallel planes. 106 Figure 6.4. Segregation energy versus dopant concentration plot for the near coincidence GB with (00.1) and (01.3) as GB parallel planes. ... 106 Figure 6.5. Atomistic structures of a Y segregated near coincidence GB at different Y-dopant concentrations, (a) 0.60 at./nm2, (b) 1.60 at./nm2, (c) 2.59 at./nm2, and (d) 3.59 at./nm2. All the atomistic structures are seen parallel to the GB plane. Yttrium atoms are in blue, oxygen red and aluminum atoms are in pink. ... 108 Figure 7.1. Conformations of the PAA molecule on hydroxylated alumina surface in vacuum at (a) t=0, (b) t=0.017 ns (tail configurations), (c) t=0.059 ns (loop configurations), and (d) t=0.069 ns (train configuration). Green atoms in the polymer molecule are hydroxyl O, pink are the backbone carbon atoms, blue are Na counter ions. ... 112 Figure 7.2. Unstable hydroxylated alumina surface in water, (a) hydroxyl atoms detach from the surface in MD simulations with De Leeuw water, while (b) Al atoms detach with TIP3P water. Al is green, H white and O atoms are in red. ... 113

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List of Tables

Table 3.1. Sigma values of the twin grain boundaries, surface area (A) of the cells used for the calculations and interfacial energies of grain boundaries (ϒGB) and

surfaces (ϒsurf) calculated in the present work. Sigma values is the ratio of the lattice

points in the unit cell of coincident site lattice and the original lattice. ... 41 Table 3.2. Asymptotic/local minimum Segregation energy values (ΔHseg) and corresponding concentration (C) for La-Y Codoped surfaces and grain boundaries.

$_{Asymptotic ΔH}

seg and C, *Value of ΔHseg and C at a local minima; #neither asymptotic

value nor minima is observed; ¥change in the slope of the energy curve ... 48 Table 3.3. Segregation energy values (ΔHseg) and corresponding concentration (C1

and C2) for Mg-La Codoped surfaces and grain boundaries. C1 and C2 are the interface

dopant concentrations when 4 and 8 dopant atoms are put on the interface respectively. ... 49 Table 3.4. Segregation energy values (ΔHseg) and corresponding concentration (C1

and C2) for Mg-Y Codoped surfaces and grain boundaries. C1 and C2 are the interface

dopant concentrations when 4 and 8 dopant atoms are put on the interface respectively ... 51 Table 3.5. Packing fractions of the single and codoped grain boundaries (Grain boundary width for packing fraction calculation was taken 10 A0 for La-Y, La and Y doping, while 6 A0 for Mg-Y, Mg-La and Mg doped grain boundaries). ... 56 Table 3.6. Dopant-Oxygen coordination number (CN) and nearest neighbor (NN) distance for different codoping combinations ... 63 Table 4.1. Results of the Al-Cl Buckingham potential parameter optimization. The experimental and the fitted lattice parameter along with the error are listed in the table. ... 69 Table 4.2. Chlorine segregation energies are listed for alumina surfaces and grain boundaries at the concentration where the slope changes in the segregation energy plot. Corresponding characteristic concentrations are also given for each interface.72 Table 4.3. Cl-Al coordination numbers of different interfaces are given with corresponding nearest neighbor distances. ... 76 Table 5.1. Details of the three primary classes of diffusive jumps of oxygen in alumina ... 87 Table 6.1. Concentrations of the Y and La dopants at three grain boundaries determined using EDX line scan analysis. It also gives the parallel planes of the adjacent grains and the misorientation angle of the GBs. ... 104

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Table 6.2. Summary of the ΔHseg at experimentally observed concentration and

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1

Introduction

This chapter sets the background for the present thesis. In this chapter we first discuss the theory of light transmission in polycrystalline alumina and then outline the key microstructural requirements to manufacture highly transparent polycrystalline alumina. The developments in the processing of transparent alumina in the last few decades are detailed out next. We identify the key processing issues, better fundamental understanding of which will help bridge the gap between knowledge and practice of processing of transparent alumina and lead to a knowledge based development of transparent alumina ceramics. At last, we state the goals and objectives of the present thesis.

1.1. Transparent Polycrystalline Alumina

Sapphire has excellent physical properties such as high strength, high temperature chemical inertness, and high degree of transparency. But high production costs and limitations in shapes and sizes prevent its use for wider potential applications such as wave guides, armor windows, high temperature refractory windows, watches, jewelry etc. On the other hand, polycrystalline alumina (PCA) has excellent mechanical properties and flexibility in shapes and sizes, but can’t be used for transparent applications due to its inferior radiant energy transmission. The quest for combining the properties of both, led Coble to discover translucent alumina in 1962 using the hot isostatic pressing (HIP) technique [1]. Since then several attempts have been made in this direction. A breakthrough was claimed when Krell et al. [2] and Apetz and Bruggen [3] reported real inline transmittance (RIT) values as high as 70% in isolated attempts using slip casting and HIP. But no follow up commercial development on these promising results leads to the suspicion that either the processing method is not robust enough or it is too expensive. In the last decade, the spark plasma sintering (SPS) technique has been investigated in order to make the processing of alumina simpler and reproducible [4,5,6,7]. Roussel et al. [4] and Stuer

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_totd) ss due to th is the thic n be defined 55% RIT v But the d umina still w

sion in P

d the light t he structur ources of lig d pores (Fig undaries an light which e Lambert B he reflectio ckness of t d as the su values resp ream of c waits to see

Polycrys

transmissio al requirem ght scatterin gure 1.1). nd pores ar passes the Beer Law: n (14% for the sample m of the gr ectively, us commercial e the light o

stalline

on mechanis ments to ach ng in a polyc re the sourc e material alumina), n , and ϒtot rain bounda sing standa scale app of day.

Alumin

sm in a po hieve high R crystalline m ces of light without ad n is the refra is the tota ary and por

ard SPS and plication o

na

lycrystalline RIT values in material are scattering a dsorption o Eq.1.1 Eq.1.2 Eq.1.3 active index l scattering e scattering d f e n e a r x g g

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3

tot pore gb

γ =γ +γ Eq.1.4

The scattering contribution by pores was described by Peelen & Matselaar [8] as:

, 2 3 4 exp(3.5 ) p eff pore Peelen m V Q r

γ

σ

= Eq.1.5

Where Vp is the specific pore volume, rm is the radius of the mode of the volume

pore size distribution, σ is the standard deviation factor for a lognormal distribution of pores, and Qeff is the effective dimensionless scattering efficiency of the pores

defined as:

( ) ( )

( )

1 2 2 0 0 eff sca Q Q r f r r dr f r r dr − ∞ ∞    =    







 Eq.1.6

Where Qsca is the scattering efficiency factor given by Mie theory and f(r) is the log

normal pore size distribution.

Apetz and Bruggen [3] derived a numerical expression for the grain boundary scattering coefficient approximating polycrystalline alumina with randomly distributed mono-dispersed spherical grains in a homogeneous matrix, which is given as: 2 2 , 2 0 3 gb Apetz r n

π

γ

λ

= Δ Eq.1.7

Where 2r is the grain size, λ0 is the wavelength of the incident light, and Δn is the

mean absolute difference in the refractive index of the sample.

As can be seen from Eq.1.5 and Eq.1.7, the microstructural requirements for higher

RIT are: ultrahigh density, small grain sizes, grain alignment, and minimum pore volume.

Above mentioned models have been used to numerically predict the RIT values of the transparent ceramics for several years. However, Stuer et al. [9] did an extensive experimental study of transparent alumina to link the observed transparency with measured porosity and defect size. They found that the previously developed models [3,8] overestimate the pore scattering. They used their experimental data to show that a more recent model developed by Pecharrroman et al. [10] explained the experimental data better than the previously developed models. The Pecharroman scattering coefficients are given as follows,

2 2 , 2 0 6 ( ) g gb Pechorroman n α ξ α γ λ Π = Δ Eq.1.8

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4 2 4 3 ₂ , 4 2 0 32 ₁ 2 p pore Pechorroman f _n n α γ λ Π  ₋  =  ₊    Eq.1.9

Where, α ξ( )being a texture function depending on the preferential texture direction of the sample, f being the pore volume fraction and n the average refractive index of the sample. α_g and α_p are the characteristic pore and grain radii, respectively, given as,

4 , , 3 , g i i g g i i i _i g i f α α α α =



=



Eq.1.10 6 , 3 3 , p i i p p i i α α α =



_

Eq.1.11

Where, αg i, and αp i, are the grain and pore radius, respectively, of the i

th

class of the size distribution, fi is the fraction of the grain with radiusαg i, . In the Pechorroman

model, the expression for grain boundary scattering coefficient is very similar to earlier proposed by Apetz as mentioned in Eq.1.7 except the separation of texturing function from nΔ . However, the pore scattering coefficient is significantly different from the one proposed by Peelen as mentioned in Eq.1.5. Stuer et al. [9] showed that with the new definition of characteristic pore size, the pores smaller than certain pore size do not contribute significantly to the pore scattering coefficient. It is rather only the bigger size pores which contribute to the pore scattering. Therefore, it can be implied that smaller size pores do not hurt the transparency of the ceramic, and hence the requirement of ultrahigh density could be more flexible.

1.3. Key Processing Developments

Having identified the high density and submicron size and/or oriented grains as the requirements for the transparent alumina, we give an overview of the developments in the processing of transparent alumina over the last few decades to progress in this direction in the following section. Majority of the work has been focused on two important aspects of the processing: developing the methods for defect free green body processing and efficient sintering methods to limit the grain growth in the final stages of sintering while maintaining the high density.

1.3.1. Defect free green body processing

Uniaxial and isostatic pressing techniques were used in the initial investigations on the synthesis of transparent/translucent alumina [1]. These techniques continue to be used by some researchers even in some recent works [7,11,12]. However, Schmidt et al. [13] emphasized on the need for the processing techniques which can

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5 produce defect-free green bodies and can build defect healing mechanism during sintering for the synthesis and processing of nanoscaled ceramics. The study concluded that nanoscaled ceramic powders can be effective in separating the densification from the grain growth provided the surface interaction of the small particles can be controlled during colloidal processing. In a series of papers by Krell et al. [2,14,15], it was pointed out that defect free green bodies with homogeneous particle coordination are essential to achieve the highest density at low temperatures and thereby limit grain growth. Wet processing routes (slip/gel casting) with high solid content but low viscosity have been found to be the most effective green body processing methods to obtain homogeneous and highly dense green bodies [3,4,5,15,16]. The difficulty of controlling interactions increases as the particle sizes become smaller. Preparing a uniform stable suspension of ultrafine ceramic powders is very difficult due to the tendency of agglomeration of high specific surface area fine ceramic powders. Organic polymer molecules are often used in the wet processing techniques in order to control the stability of the colloidal suspension of alumina nano particles [17]. Short chain polymer molecules containing easily dissociable functional groups mostly provide electrostatic stabilization, while medium molecular weight polymer molecules (10000<MW<25,000) can provide steric as well as electrosteric stabilization based on solution condition and conformation of the polymer molecule [18]. Dolapix [15], ammonium polymethacrylate [2] , Carboxylic acids [13] have been used in the literature to stabilize the alumina nano particle suspensions.

1.3.2. Use of modern sintering methods and sintering aids

Until the last decade, researchers have focused on conventional sintering methods and hot isostatic pressing (HIP) to obtain fully dense transparent alumina [1,19]. Krell et al. [14,20] and Apetz and Bruggen [3] employed a pressureless pre sintering followed by HIP to obtain fully dense alumina with a grain size in the submicron range. In the recent times microwave sintering [12] as well as spark plasma sintering [4,5,6] have been successfully used to obtain fully dense transparent alumina with RIT values up to 70%. In addition to defect free green body processing and new sintering techniques, sintering additives such as, Y, La, and Mg, have also been used to control the grain growth during the final stages of sintering [4,5,21]. It has been shown in the studies focusing on creep enhancement as well as transparent ceramics that the addition of dopants results into smaller grain size. In addition, it has been observed that codoping, i.e. combination of dopants, results in smaller grains, higher RIT’s and higher creep resistance in comparison to singly doped alumina [5,22,23,24,25]. In 2003 two separate groups reported an RIT value of about 70% without using any dopants [2,3]. However, there has not been any follow up work on that so far. Since then there have been several studies on the effect of dopants on the RIT enhancement in alumina [4,5,11,19]. The best results in RIT enhancement of

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6

transparent alumina with sintering agents were reported very recently by Roussel et al. [4], who achieved an RIT value of 71% with La doping and spark plasma sintering method. Stuer et al. [5] studied the effect of single doping as well as codoping on the transparency of alumina. They reported the best RIT values with La and Y codoped alumina in comparison to singly doped Y, La or Mg.

1.4. Need and Scope of the Present Thesis

In the following section we discuss the existing gaps between knowledge and practice of the transparent alumina processing, which are essentially the topics of investigation of the present thesis. As mentioned in the previous section, the two major advancements in the processing of transparent alumina have been colloidal processing of alumina nano particles to form the green bodies and use of sintering additives to limit the grain growth during sintering. Adsorption mechanism of the dispersants which are used in the colloidal processing of alumina is not very well understood at the atomic scale. On the sintering side, better understanding of the effects of dopants segregation on the grain boundary structure and solid state oxygen diffusion mechanism in alumina could help in achieving the microstructural requirements for transparent alumina.

1.4.1. Adsorption of dispersants on Alumina Surface

Controlling the agglomeration of ultrafine powders in a colloidal suspension is a challenge in the wet processing of nano scaled ceramic powders. Carboxylic acids/polymers, e.g. acetic acid, citric acid, polyacrylic acid (PAA), have been used as the dispersants for achieving the well dispersed suspensions of ultrafine alumina powders [2,13]. Recently carboxylic dispersants have also been tested to induce the particle surface orientation and hence grain alignment in the sintered alumina under the influence of magnetic field [26]. It is known that the charged dispersants are chemisorbed on the alumina surface and thereby stabilize the suspension due to double layer interaction as well as steric hindrance between the particles [27,28]. However, adsorption of a polyelectrolyte on the powder surface is a complex process to understand experimentally due to the very small thickness of the adsorbed layer, in the range of few nanometers, and the difficulty of characterization methods for wet samples. The rheology of the suspension becomes further complex when using ultrafine powders in solutions of high ionic strength, which is a consequence of dopant addition during the colloidal processing of alumina. Adsorption behavior of dispersants on alumina surface is the deciding factor for its effectiveness, which depends on solvent conditions [29], powder surface characteristics [30], as well as conformational entropy of the polymer [31]. Experimental studies have helped to understand the overall effect of dispersants on the colloidal stability of alumina and have proposed possible hypothesis for polymer adsorption. However, due to the lack of the definitive fundamental understanding of

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7 adsorption mechanisms of polymer molecules on alumina surfaces the control on colloidal processing and its parameters is still very limited. Atomistic simulation is a promising approach to extend the understanding of the basic issues due to its approachable length scales, time efficiency and ease to control different parameters individually. The goal of this part of the work was to understand the adsorption mechanism of polymer dispersant, poly-acrylic acid, at the atomic level. Preliminary work was done on the effect of the substrate surface on the adsorption mechanism of organic molecules, dependence of the conformational arrangement of dispersants on the alumina surface characteristics. Adsorption of PAA was simulated on the characteristic, morphology dominating alumina surfaces using the molecular dynamics approach based on empirical force fields.

1.4.2. Grain Boundary Segregation of Dopants

In addition to the defect free green body processing, limiting the grain growth in the final stages of sintering is another big challenge to achieve fully dense transparent alumina with minimal grain coarsening. Various rare earth elements (La, Mg, and Y) have been employed in the past as sintering aids/dopants for the sintering of transparent alumina [4,5,11,21]. The larger dopants have very low solubility in the bulk alumina and segregate to the grain boundaries, thereby reducing the rate of densification and grain size/sintered density ratio [32,33]. Recently, codoping of alumina with rare earth elements (e.g. La-Y, Y-Mg, Mg-La) has been reported to further reduce the creep rates of alumina [24,34], as well as to increase the RIT in alumina [5]. Several propositions have been made to explain the additive effect of codoping over single doping [21,32,35]. But, the basic mechanism behind the additive effect of codoping over single doping is far from well understood and conclusive. Therefore, the second objective of the current thesis was to improve the understanding of the atomistic mechanisms behind doping and codoping using classical atomistic modeling methods. Energy minimization method based on empirical potentials was used to calculate the relaxed surfaces and grain boundaries. The effect of doping and codoping was studied on 9 surfaces and 8 grain boundaries for three codoping combinations: La–Y, La–Mg and Mg–Y. These results will give interesting insights into interfacial energies and consequent grain growth for better control of microstructures toward transparent ceramics.

1.4.3. Solid State Diffusion of Oxygen in Alumina

Knowledge of diffusion in alumina is crucial for understanding high temperature processes such as, diffusional creep, sintering of polycrystals, plastic deformation of single crystals, and alumina scale formation in Al containing alloys. Tracer diffusion experiments are conducted with O18 to determine the oxygen diffusion coefficient of alumina at different temperatures. The activation energy and pre-exponential constant are calculated using the Arrhenius equation

D D

=

0

exp

(

−

E RT

a

/

)

. The

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8

activation energy of oxygen diffusion in alumina via vacancy mechanism has been reported to be 5-6 eV in several experimental studies [36]. On the other hand, several computational methods have been used to simulate the solid state oxygen diffusion in alumina using force field as well as ab-inito methods. However, the atomistic simulation studies report the activation energy of oxygen vacancy diffusion in pure alumina to be 1-2.5 eV, not being able to capture the experimental values [37,38]. This is also popularly known as the conundrum of oxygen diffusion in alumina, a phrase coined in ref [39]. Diffusion process in undoped alumina is controlled by oxygen vacancy diffusion since the rate of diffusion of Al3+ is much faster than O2- [38,39]. As mentioned earlier, alumina is often doped with transition elements (e.g. La, Y, Mg) to enhance its mechanical and optical properties, which also affects the diffusion process. In addition to the difficulty in simulating oxygen diffusion in bulk alumina due to its multiscale nature, the presence of impurity/dopants makes it an even more complex problem to handle in atomistic simulation. The objective of this part of the work was to devise a methodology to simulate accurately and efficiently multiscale problem of oxygen diffusion in bulk alumina. In the present work, nudged elastic band method [40] based on empirical force field was used to study the effect of Mg on the oxygen diffusion in alumina. Binding energies and migration barriers of diffusive jumps in the neighborhood of Mg were calculated to gain an insight into the defect cluster formation, migration pathways, and defect destabilization. Understanding of diffusion mechanism and effect of Mg on diffusion in alumina will be very useful to understand the mass transport and grain growth process during the sintering of transparent alumina and in turn provide a better control over the doping process of alumina.

1.5. Objectives

To summarize what has been discussed so far, polycrystalline alumina with submicron size grains, high density and oriented grains needs to be synthesized. Polymer dispersants are used to form stable colloidal suspension of alumina nano powders. But their efficiency and utility can be enhanced with the better understanding of their adsorption mechanism on particle surfaces. Further insights into the surface specific adsorption behavior of polymer additives at the atomic level needs to be gained using the atomistic simulation methods, which could help forming green bodies with controlled particle orientation, and sintered piece with desired grain orientation thereafter. On the sintering aspect, the grain growth should be minimized in the final stages of sintering while ensuring the high density of alumina. The doping of alumina has been shown to be effective in limiting the grain growth. However, it seems to have reached a stagnation phase, where no more tangible benefits are possible with the same doping methods. Better fundamental atomic level understanding of the effects of dopants type, concentration and doping

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9 strategy (single/co/multiple doping) on the grain boundary structure is required to devise new doping methods to increase the efficiency of doping in limiting the grain growth and controlling the microstructure. Finally, the discrepancy in the activation energy of the solid state oxygen diffusion between simulation and experiments calls for taking into account the real system phenomena e.g. effects of dopants/impurities on the oxygen diffusion in alumina. Resolution of this issue will not only enhance our understanding of high temperature sintering of alumina ceramics but many high temperature processes, e.g. diffusional creep, sintering of ceramics, plastic deformation of single crystals and alumina scale formation in Al containing alloys.

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10

Computational Methods

This chapter gives the brief description of the computational methods used in the current work. Full details of these methods are out of the scope of the present work, but the following chapter should serve well as an introduction to the most commonly used atomistic simulation methods and the references provided can be followed for a deeper understanding. All the methods used in the present work are based on the classical description of energy via empirically derived analytical functions and fitting parameters. The chapter also gives the details of the construction of the different types of simulation cells, boundary conditions, and the limitations of the methods used in the present work.

2.1. Force Field Description

To start with the atomistic simulations, the first basic requirement is to have an accurate force field. The accuracy of any force field based simulation depends heavily on the accuracy of the force field. A force field is essentially a mathematical function used to describe the potential energy of the system in combination with the parameters to fit the mathematical form for various atom types present in the system. The entire force field approach is based on the Born-Oppenheimer approximation that the effect of electrons can be embodied into a single potential energy function which depends only on the nuclear positions. The potential energy of a system can be described as a combination of the bonded and non-bonded interactions (Eq.2.1). Bonded interaction depends on angles and torsion as well as the chemical bond (Eq.2.2). Non-bonded interaction consists of long range electrostatic interaction and short range van der Walls interactions (Eq.2.3).

total bonded non bonded

E

=

E

+

E

₋ Eq.2.1

bonded bond angle dihedral

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11 non bonded electrostatic vanderWalls

E

₋

₌

E

₊

E

Eq.2.3

2.1.2. Bonded interaction

The force between the atoms which are connected by covalent bonds is represented by bonded potentials. Bonded potential consists of mainly three components: Bond energy, bond angle energy, and bond torsion energy.

Bond potential is the energy associated with the stretching of the bonds. It is represented in the form of a spring potential. The energy of the bond is,

2 0

(

)

bond harmonic l

E

₌

k r r

₋

_Eq.2.4

kl is the spring constant for the bond potential and r0 is the equilibrium bond

distances. This means that the bond energy increase if the bond stretched or compressed, and keeps on increasing as the bond is stretched further as shown in the Figure 2.1. However, in the real molecules, the bond will break if the bond is stretched sufficiently, which is not predicted in this approach. However this model fits well around the equilibrium bond length. Therefore, other mathematical forms have been developed to fit better to the real molecular bonds. The Morse potential is one of the very commonly used amongst them. It is given as,

(

1

( 0)

)

2

1

bond r r morse

E

₌

D



_

₋

e

−α −

₋



_





Eq.2.5

Where, D is the dissociation energy of the bond and α is a parameter to describe the anharmonicity of the bond. Figure 2.1 shows that the Morse potential captures the anharmonicity of the bond well and hence can be applied in wider situations than the simple harmonic potential.

Energy associated with the bending of the bond angles from their equilibrium position is again modeled using the harmonic approximation,

2 0

(

)

angle

E

=

k

_θ

θ θ

−

Eq.2.6

kƟ is the spring constant for the bond angle potential, Ɵ and Ɵ0 are the bond angle

and equilibrium bond angles, respectively. Once again, the harmonic approximation for the bond angle potential fits best close to the equilibrium bond angle. But, it is more adequate to use in this case since the bond angles do not change much.

Energy associated with the torsion (rotation around the bond axis) of the bond is represented by a periodic cosine function for the torsion potentials,

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12 torsion

E

kф is th to desc Figure 2 their fu potenti

2.1.3.

Non-bo range v

(

1

n

=

k

φ

+

e force con ribe the nu 2.1. Schema unction for al

Non-bon

onded inter van der Wal

(

cos n

φ

+

−

nstant to tw mber of osc atic diagram rm, (a) bo

nded inte

actions are ls interactio

)

0

φ

−

wist bond ar cillations wi m of variou nded pote

eractions

e the sum o ons. round the b ithin one co s types of b ential, (b) a

s

of long rang bond axis, n omplete rev bonded inte angle pote ge electrost is a scaling volution. eraction pot ential, and tatic energy Eq.2.7 g paramete tentials and (c) torsion y and short r d n t

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13

2.1.3.1. Electrostatic interaction

Electrostatic potential energy between two charged ions is given by,

, 0

4

i j ij electrostatic ij

q q

E

r

πε

=

_Eq.2.8

qi and qj are the charges on ions i and j respectively, and ϵ0 is the permittivity of

vacuum. The electrostatic potential energy decays as the inverse power of r, but the number of interacting ions increases as the power of r2. Hence, the energy density increases with the distance instead of decreasing, which makes it difficult to integrate the electrostatic interaction energy over all the pairs of ions using a standard integration mechanism.

The most commonly used method to address this problem is the Ewald summation method [41]. It first applies a Laplace transformation to the coulomb term to accelerate the evaluation and then is separated into two terms. One of these terms converges quickly in the real space and the second decays quickly in the reciprocal space. The self-energy term is subtracted in order to evaluate the interaction correctly.

Coulomb real reciprocal self

E

₌

E

₊

E

₊

E

_Eq.2.9 1 2 1 1

1

2

N N i j real ij i j ij

q q

E

erfc

r

η

= =





=











Eq.2.10

(

)

2 2 1 1

exp

4

1

4 exp

.

2

N N reciprocal i j ij i j G

G

E

q q

iG r

V

G

η

π

= =





−









=



_Eq.2.11 1 2 2 1 N self i i

E

q

η

π

=

 

= −



 

_{ }

Eq.2.12

Here q is the charge on an ion, G is the reciprocal lattice vector, V is the volume of the unit cell, and η is the parameter which divides the work between real and reciprocal space. The cut off distance (Rmax and Gmax) is applied in calculating both

real as well reciprocal terms. One of the criteria to optimize the values of Rmax, η and

Gmax is to minimize the number of terms in both summations. For a given accuracy A,

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14 1 3 ₃ opt

Nw

V

π

η

= 











Eq.2.13 1 2 max

ln A

r

η



₋



= 







Eq.2.14

(

)

1 ₁ 2 ₂ max

2 ln

G

₌

_η

₋

A

_Eq.2.15

w is the parameter which defines the relative computational cost of real and

reciprocal term calculation.

2.1.3.2. Short Range Interactions

Short range interaction becomes important when the atoms are in the immediate coordination shells. Short range attractive term is an interaction due to the instantaneous dipole- instantaneous dipole interaction energy which comes from quantum mechanics and is represented by the r6 inverse term. The repulsive term is introduced in order to avoid the electron cloud overlapping. For the ionic cases, it is either represented by a positive term which varies inversely with distance, or an exponential form. The first one is called the Lenard-Jones potential (Figure 2.2b) and the second one is called the Buckingham potential (Figure 2.2a),

6 6

exp

ij buckingham ij

r

C

E

A

r

ρ





=



−



−





Eq.2.16 6 6 m Lenard Jones m ij ij

C

E

r

−

=

−

Eq.2.17

A, ρ, C6 and Cm are the parameters which are fitted with experimental and/or

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Figure 2.2. (a) Bucking Jones pote 2.1.3.3. Po Polarisabili Dick and O Functional gham poten ntial olarisabilit ty of atoms Overhauser form of tw ntial for Al-O

ty s can be tak [42]. In thi (a) (b) o main type O and O-O

ken into acc s model a m Atomist es of non-bo pairs, and count by a massless sh tic Modeling o onded dispe (b) typical core-shell m hell is attac of Transparen ersion inter shape of a model prop ched to the t Alumina 15 ractions, Lenard-posed by core of

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16

the atom with a spring. The core consists of the nucleus and the inner shell of the electrons, while the shell consists of the valence electrons. The potential energy between the core and the shell is given by a harmonic spring potential,

2

1

2

spring c s c s

E

₌

k r

₋ ₋ _Eq.2.18

kc-s represents the rigidity of the atom and rc-s is the distance between the core and

the shell. Conventionally, short range forces act only on the core, while the electrostatic force acts on the core as well as the shell. Due to this discrepancy in the forces on the core and the shell, the environment also plays a role in the polarisability of the atoms. While performing molecular dynamics (section 2.4) with core-shell model, shells are needed to be given special treatment because the approach fails with the massless shells. The problem is addressed either by assigning a small mass to the shells allowing the shells to follow the normal Newton’s laws of motion. Another approach is to treat shells adiabatically, where the shell positions are optimized after every time step. However, this model takes into account only the dipole-dipole interaction into account. There can be higher orders of ion distortions which can be important in certain cases, especially in high symmetry structures.

2.1.4. Force Filed Optimization

To develop the force field for a material or for a particular modeling problem, first task is to identify the right functional form to correctly represent the potential energy of the function, which depends mostly on the material characteristics. Once the functional form is chosen, the parameters of the functional form need to be optimized. The optimization procedure depends on the desired accuracy and goal of the modeling task. Direct transfer of the known parameters from one system to another similar system by analogy is the minimal form of optimization. It is fast and easy. E.g. If the force field for alumina is known and we want to simulate the grain boundary segregation of Cl, but we don’t know the interaction potential parameters for Al-Cl. Keeping rest of the potential parameters constant, only the Al-Cl interaction parameters can be fitted to the lattice properties of AlCl3, a compound

which has both Al and Cl and whose experimental properties or abinito calculations are available in the literature. Properties to be fitted vary depending on the goal of the simulation task.

On the other hand, in a maximal approach the parameters of the force field are derived from the scratch. It is much more time consuming and requires appropriate target data to be fitted with, but it is more accurate and precise for the goal of the calculation. Not only the parameters of a functional form have to be optimized, but also the process of optimization starts with identifying the functional form itself.

Atomistic Modeling of Processing Issues for Transparent Polycrystalline Alumina